Symbolic power of an ideal

In algebra and algebraic geometry, given a commutative Noetherian ring ${\displaystyle R}$ and an ideal ${\displaystyle I}$ in it, the n-th symbolic power of ${\displaystyle I}$ is the ideal

${\displaystyle I^{(n)}=\bigcap _{P\in \operatorname {Ass} (R/I)}\varphi _{P}^{-1}(I^{n}R_{P})}$

where ${\displaystyle R_{P}}$ is the localization of ${\displaystyle R}$ at ${\displaystyle P}$, we set ${\displaystyle \varphi _{P}:R\to R_{P}}$ is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of ${\displaystyle R/I}$.

Though this definition does not require ${\displaystyle I}$ to be prime, this assumption is often worked with because in the case of a prime ideal, the symbolic power can be equivalently defined as the ${\displaystyle I}$-primary component of ${\displaystyle I^{n}}$. Very roughly, it consists of functions with zeros of order n along the variety defined by ${\displaystyle I}$. We have: ${\displaystyle I^{(1)}=I}$ and if ${\displaystyle I}$ is a maximal ideal, then ${\displaystyle I^{(n)}=I^{n}}$.

Symbolic powers induce the following chain of ideals:

${\displaystyle I^{(0)}=R\supset I=I^{(1)}\supset I^{(2)}\supset I^{(3)}\supset I^{(4)}\supset \cdots }$